hw1 - Math W4051: Problem Set 1 due Wednesday, September 12...

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Unformatted text preview: Math W4051: Problem Set 1 due Wednesday, September 12 1. Let d 1 and d 2 be two metrics on the same set X with the property that there exist constants C, C > 0 such that C d 1 ( x, y ) d 2 ( x, y ) C d 1 ( x, y ) , for all x, y X. (a) Show that a sequence { x n } of elements of X converges to some x X in the metric d 1 if and only if it converges in the metric d 2 . (b) Show that a subset U X is open with respect to d 1 if and only if it is open with respect to d 2 . 2. Let d be a metric on X and set d ( x, y ) = d ( x, y ) 1 + d ( x, y ) . (a) Show that d defines a metric on X. (Hint: If f ( x ) = x/ (1 + x ) for x > , use the mean-value theorem to show that f ( a + b )- f ( b ) f ( a ) for a b. ) (b) Show that a sequence { x n } of elements of X converges to some x X in the metric d if and only if it converges to x in the metric d . 3. Let X be the set whose elements are the sequences { x n } = ( x 1 , x 2 , . . . ) of real numbers....
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hw1 - Math W4051: Problem Set 1 due Wednesday, September 12...

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